After I unlock a KDE session, the small borderless window that collected the user's password remains on screen, on all desktops, above all other windows. This happens since KDE 4.9.0 (I'm now on 4.9.1) and only when the screen has been locked for at least 12 hours or so. Or maybe it's just the date change that triggers this behaviour.

I have had the same problem with KDE 4.9.1, but it is only after my laptop has been locked for several hours. I didn't have the problem with 4.9.0 but I'm not sure what the exact reason for the problem is.

Can you please list everything you've installed together with KDE 4.9.1? Using genlop or anything similar.
Your statement makes me think the culprit is not KDE itself, but some supporting library, like mesa-8.1_rc1_pre20120814. My video driver is xf86-video-intel-2.20.6.

The version of mesa that I have installed is mesa-9_pre20120831-r1:0 and my video driver is xf86-video-intel-2.20.7. It is also probable worth noting that I also have the kde use flag enabled on a global scale.

If you are looking for any other information, let me know and I can list it here.

I have this problem as well, I guess since 4.9.0.
The 'dialog window' do disappear for me if I suspend the desktop effects (ALT+SHIFT+F12) and re-enable them (ALT+SHIFT+F12).
My uneducated guess is that this issue is in the composition engine of KDE, but I know nothing about Window managers

I have this problem as well... doesn't happen every time but it does happen a lot. So far my workaround is to
[..]
Also I have not had ANY similar issues with any other dialogs or screen artifacts whatsoever -- only the unlock dialog.